Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos

Abstract

We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. In particular, we show that chaotic random variables enjoy the following form of universality: (a) the normal and chi-square approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums.

Published in at http://dx.doi.org/10.1214/10-AOP531 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Central limit theorems for sequences of multiple stochastic integrals
• Normal approximation under local dependence
• Noncentral convergence of multiple integrals
• Random matrices: Universality of ESDs and the circular law
• Moments, cumulants and diagram formulae for non-linear functionals of random measures
• Universal Gaussian fluctuations of non-Hermitian matrix ensembles

Cited by in corpus (38)

• Chaos of a Markov operator and the fourth moment condition
• Wigner chaos and the fourth moment
• Fourth moment theorems on the Poisson space in any dimension
• Ergodicity and Gaussianity for Spherical Random Fields
• Asymptotic behavior of large Gaussian correlated Wishart matrices
• Invariance principles for homogeneous sums of free random variables
• Total variation distance between stochastic polynomials and invariance principles
• Motif Estimation via Subgraph Sampling: The Fourth Moment Phenomenon
• A Peccati-Tudor type theorem for Rademacher chaoses
• Edwards-Wilkinson fluctuations for the directed polymer in the full $L^2$-regime for dimensions $d\ge 3$
• Convergence of densities of some functionals of Gaussian processes
• The impact of the diagonals of polynomial forms on limit theorems with long memory
• Gaussian Limits for Subcritical Chaos
• Lectures on Gaussian approximations with Malliavin calculus
• New Berry-Esseen bounds for non-linear functionals of Poisson random measures
• Optimal Convergence Rates and One-Term Edgeworth Expansions for Multidimensional Functionals of Gaussian Fields
• Scaling limits of disordered systems and disorder relevance
• Homoscedasticity tests for both low and high-dimensional fixed design regressions
• Malliavin-Stein method for Variance-Gamma approximation on Wiener space
• On the fourth moment condition for Rademacher chaos
• Entropy and the fourth moment phenomenon
• The multivariate functional de Jong CLT
• Malliavin-Stein Method: a Survey of Recent Developments
• Berry-Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences
• Universality of free homogeneous sums in every dimension
• The universality of homogeneous polynomial forms and critical limits
• On the Rate of Convergence to a Gamma Distribution on Wiener Space
• Fluctuations of Quadratic Chaos
• Gaussian fluctuations from the 2D KPZ equation
• The optimal fourth moment theorem
• Gaussian approximation of maxima of Wiener functionals and its application to high-frequency data
• An approximate randomization test for high-dimensional two-sample Behrens-Fisher problem under arbitrary covariances
• Berry-Esseen bounds for the multivariate $\mathcal{B}$-free CLT and operator-valued matrices
• Hypercontractivity Meets Random Convex Hulls: Analysis of Randomized Multivariate Cubatures
• Multidimensional limit theorems for homogeneous sums: a general transfer principle
• Malliavin calculus for marked binomial processes: portfolio optimisation in the trinomial model and compound Poisson approximation
• A Moment Majorization principle for random matrix ensembles
• Quantitative CLTs for symmetric $U$-statistics using contractions